Question:medium
A modified Taylor tool life equation is given as follows:
\[ V T^n f^m = constant \]
where \( V \) is the cutting speed (m/s), \( T \) is the tool life in minutes and \( f \) is the feed in mm/rev, \( n = 0.25 \) and \( m = 0.5 \). Under two different cutting conditions \( (V_1, f_1) \) and \( (V_2, f_2) \) the tool life \( (T_1, T_2) \) was found to be the same.
If the ratio of the cutting speeds \( (V_1/V_2) \) used is 2/3, then the ratio of corresponding feeds \( (f_1/f_2) \) must be ______ (rounded off to two decimal places).
A modified Taylor tool life equation is given as follows:
\[ V T^n f^m = constant \]
where \( V \) is the cutting speed (m/s), \( T \) is the tool life in minutes and \( f \) is the feed in mm/rev, \( n = 0.25 \) and \( m = 0.5 \). Under two different cutting conditions \( (V_1, f_1) \) and \( (V_2, f_2) \) the tool life \( (T_1, T_2) \) was found to be the same.
If the ratio of the cutting speeds \( (V_1/V_2) \) used is 2/3, then the ratio of corresponding feeds \( (f_1/f_2) \) must be ______ (rounded off to two decimal places).
\[ V T^n f^m = constant \]
where \( V \) is the cutting speed (m/s), \( T \) is the tool life in minutes and \( f \) is the feed in mm/rev, \( n = 0.25 \) and \( m = 0.5 \). Under two different cutting conditions \( (V_1, f_1) \) and \( (V_2, f_2) \) the tool life \( (T_1, T_2) \) was found to be the same.
If the ratio of the cutting speeds \( (V_1/V_2) \) used is 2/3, then the ratio of corresponding feeds \( (f_1/f_2) \) must be ______ (rounded off to two decimal places).
Show Hint
Cancel the equal tool-life term \( T^n \) from both conditions, then solve \( (f_1/f_2)^m = V_2/V_1 \) for the feed ratio.
Updated On: Aug 5, 2026
Show Solution
Correct Answer: 2.25
Solution and Explanation
Step 1: Understanding the Concept:
In the modified Taylor equation $VT^n f^m = constant$, the tool life stays the same in both cutting conditions given in the problem.
That means the $T^n$ part is common to both sides and drops out of the comparison.
We are left with a direct relation between cutting speed and feed that we can solve for the feed ratio.
Step 2: Key Formula or Approach:
Equating the two conditions and cancelling $T^n$:
\[ V_1 f_1^m = V_2 f_2^m \]
\[ \left( \frac{f_1}{f_2} \right)^m = \frac{V_2}{V_1} \]
Step 3: Detailed Explanation:
We are given $m = 0.5$ and $V_1/V_2 = 2/3$, so $V_2/V_1 = 3/2 = 1.5$.
Substituting:
\[ \left( \frac{f_1}{f_2} \right)^{0.5} = 1.5 \]
Since the power on the left is 0.5 (a square root), we remove it by squaring both sides:
\[ \frac{f_1}{f_2} = (1.5)^2 = 2.25 \]
Final Answer:
Squaring both sides gives the feed ratio directly. \[ \boxed{2.25} \]
In the modified Taylor equation $VT^n f^m = constant$, the tool life stays the same in both cutting conditions given in the problem.
That means the $T^n$ part is common to both sides and drops out of the comparison.
We are left with a direct relation between cutting speed and feed that we can solve for the feed ratio.
Step 2: Key Formula or Approach:
Equating the two conditions and cancelling $T^n$:
\[ V_1 f_1^m = V_2 f_2^m \]
\[ \left( \frac{f_1}{f_2} \right)^m = \frac{V_2}{V_1} \]
Step 3: Detailed Explanation:
We are given $m = 0.5$ and $V_1/V_2 = 2/3$, so $V_2/V_1 = 3/2 = 1.5$.
Substituting:
\[ \left( \frac{f_1}{f_2} \right)^{0.5} = 1.5 \]
Since the power on the left is 0.5 (a square root), we remove it by squaring both sides:
\[ \frac{f_1}{f_2} = (1.5)^2 = 2.25 \]
Final Answer:
Squaring both sides gives the feed ratio directly. \[ \boxed{2.25} \]
Was this answer helpful?
0
